Download A Guide to Vectors 2 Dimensions

A Guide to Vectors 2 Dimensions
Teaching Approach
For Grade 11, it will help to begin with concepts of displacement and velocity that the
students may have done in Grade 10. Videos 1 and 2 will remind them how vectors can be
added, and what a resultant vector is.
CAPS for Grade 11 emphasises force as a vector quantity; this can help you because you
can ask students to push or pull objects and they can sense in their arms that force has both
magnitude and direction, and most importantly, that two forces in different directions can
result in a force in a new direction. For example, if two students pull a desk in directions that
are at right angles to each other, the desk will move in a direction that is in-between the two
directions they are pulling. They can watch this principle at work as they use the forces
board in Lesson 3: two strings pulling sideways in different directions produce a resultant
force that acts vertically upward.
The videos have only a few questions in them so make use of the task video to ensure that
the students exercise the ideas you are teaching them through the Mindset lesson videos.
Where the video shows a PAUSE sign, stop the video. Repeat the question for the class,
pause the video at the relevant graphic, and give them enough time to work on the question.
By “enough time”, we mean about 5 to 10 minutes. This could be time enough for them to
draw sketches and discuss the problem with a partner.
Learning difficulties and misconceptions
1. In Lesson 3, you will need a forces board, similar to
the one you see here. You fix a sheet of paper to the
board and the students trace the position and
direction of the strings by drawing on the paper.
But the lengths of the strings have nothing to do with
the tension forces in the strings. The tension force is
shown by the length of the vector arrow that
represents the force. So a short string may have
large force on it and the long string may have a small
force on it. Only the length of the arrow matters.
2. After you have marked the positions of the strings,
and drawn the lengths of the vector arrows to
represent the forces, you may find that your vectors
don’t quite meet up to form a closed polygon. This is
probably due to some hidden friction force that prevents the strings moving quite freely.
Check that the pulley-wheels spin easily, and that the weights don’t rub against the
board.
3. Students find it strange that we can represent forces by vectors — arrows of a certain
length and direction — and then treat them as if we were doing geometry and
trigonometry. They will get used to the idea after doing a few diagrams and calculations.
But watch for the following source of confusion: in geometry we
label the vertices (the corners) of a triangle with capital letters, so we
write triangle ABC, and we can write about “side AB”. When working
with vectors, the whole vector gets one label, such as vector F1. In
this diagram, F1, F2 and F3 are three vectors, not the vertices of the
triangle. So we could not write “side F1F2 “because it would have no
meaning.
Video Summaries
Some videos have a ‘PAUSE’ moment, at which point the teacher or learner can choose to
pause the video and try to answer the question posed or calculate the answer to the problem
under discussion. Once the video starts again, the answer to the question or the right answer
to the calculation is given
Mindset suggests a number of ways to use the video lessons. These include:
 Watch or show a lesson as an introduction to a lesson
 Watch or show a lesson after a lesson, as a summary or as a way of adding in some
interesting real-life applications or practical aspects
 Design a worksheet or set of questions about one video lesson. Then ask learners to
watch a video related to the lesson and to complete the worksheet or questions, either in
groups or individually
 Worksheets and questions based on video lessons can be used as short assessments or
exercises
 Ask learners to watch a particular video lesson for homework (in the school library or on
the website, depending on how the material is available) as preparation for the next day’s
lesson; if desired, learners can be given specific questions to answer in preparation for
the next day’s lesson
1. Relative Velocity Vectors
The lesson recaps Grade 10 vectors in one dimension, and then introduces motion in two
dimensions. A boat full of children is crossing a river. The boat has a velocity but the river
has its own velocity. Where does the boat end up?
2. Vectors of Relative Motion
In lesson 2 the boatman finds a way to steer the boat so that its resultant velocity is
straight across the river. Students use Pythagoras’s theorem and trig ratios to calculate
the magnitude of the resultant vector.
3. Forces in Equilibrium
This lesson looks at the classic demonstration of forces in equilibrium, using a forces
board. The concept of the resultant force and the polygon of forces emerge from the
activity. Again, the students have to deal with the addition of vectors.
4. Calculating Vectors using Pythagoras
In this lesson the students see that while they must add vectors to find a resultant vector,
when the vectors form a right-angled triangle, then Pythagoras can be used to calculate
the magnitudes of the vectors.
5. Resolution of Vectors into Components
Resolve a vector into its component vectors in a right-angled co-ordinate system. If the
angle ɵ between the vector R and one of the axes is known, the component vectors are
Rcosɵ along the x-axis and Rsinɵ along the y-axis.
Resource Material
1. Relative Velocity Vectors
2. Vectors of Relative Motion
3. Forces in Equilibrium
4. Calculating Vectors using
Pythagoras
5. Resolution of Vectors into
Components
http://cnx.org/content/m14030/latest
/
A resource on relative velocity in two
dimensions.
http://www.examsolutions.net/math
srevision/mechanics/vectors/relativevelocity/tutorial-1.php
A video introducing relative velocity.
https://www.youtube.com/watch?v=
nxF47gTE2yU
This video discusses relative motion.
http://www.physicstutorials.org/hom
e/mechanics/1d-kinematics/relativemotion
Relative motion with examples.
http://fiziknota.blogspot.com/2009/0
5/analysing-forces-inequilibrium.html
Analyzing forces in equilibrium.
https://www.uwstout.edu/physics/up
load/Force-Equilibrium.pdf
A laboratory activity demonstrating
the conditions under which an object
is held in static equilibrium due to
concurrent forces.
http://www.physics.arizona.edu/phy
sics/gdresources/documents/03_Ve
ctors.pdf
A study of vectors in the context of
force
http://www.mathsisfun.com/algebra/
vectors.html
All about calculating vectors.
http://www.wikihow.com/Resolve-aVector-Into-Components
How to resolve
components.
a
vector
into
Task
Question 1
Question one. You remember the aeroplane that accelerated and climbed?
aeroplane again . . . look at the sketch.
The thrust vector is 2 000 N and the lift vector is 500 N.
Here’s the
Do a scale drawing of the vectors and measure the resultant R and the angle of climb ɵ. In
your drawing, use a scale of 1 cm to 200 N. You’ll need a protractor to measure the angle ɵ.
Question 2
2.1 Work out the resultant force on the aeroplane using the theorem of Pythagoras.
2.2 Pythagoras does not work for all triangles. Why can you use Pythagoras on this
triangle?
Question 3
Here is the forces board again, and you can see a hand holding the
[Use task video Fig 2]
middle weight up.
The weights on the three strings are each 3N. When the weight
in the middle can hang freely, in what directions will the strings
lie?
3.1 First do a rough sketch of the strings as you think they will
lie when the forces are acting on them
3.2 Sketch the force vectors acting at the knot in the middle.
What do you know about these three vectors?
3.3 The angles between the strings are equal. Think of a reason
why that is so.
3.4 Now do a scale drawing of the vectors, using a scale of 3 cm
to 1 N.
3.5 Draw the resultant that will balance the force that acts vertically downward.
3.6 Draw the resultant and two other vectors head to tail to show how the two vectors
together cause the resultant.
3.7 Draw the polygon of forces that act around the knot. In this case, the polygon has three
sides.
3.8 What are the angles between the vectors?
Question 4
In this question you must explain why wet washing on a line can pull down the poles of the
washing line. Look at the diagram.
The line is made of wire, that won’t stretch. When there
is no washing on the wire, the wire is almost straight.
But now look at this diagram [below]. . . when a wet
towel that weighs 20 N is hung in the middle of the line,
the poles bend a little and the wire makes an angle of 10 ° with the horizontal.
Vector T is the tension in the wire. T has two component vectors, a vertical component of 10
N and a horizontal component that you must work out. The vertical downward component of
10 N is caused by one half of the weight of the towel. The other half of the towel’s weight
causes another vertical component at the other end of the wire
4.1 First of all, sketch the situation for yourself. On your sketch fill in all the information you
have about this system.
4.2 What is the horizontal component of the tension in the wire? Here’s a hint . . . if you can
find a right-angled triangle in your sketch, then you can use a trig function. Try the sine
or tan functions. You’ll need a calculator.
4.3 Compare the magnitude of the horizontal force that tries to bend the pole with the 10 N
vertical force that helps to hold the towel up.
4.4 Use the same method to work this out: if you made the wire tighter, so that it sagged
only half as much, that is, only 5 °, would you have to double the horizontal force?
4.5 Work out the tension T in the wire when the wire sags at an angle of 10 ° and the wire
must hold up 10 N of the weight of the wet towel. You will need a different trig function
for this.
Task Answers
Question 1
You should be able to find that the angle is 14°, measuring with your protractor.
Make sure that the angle between vector L and vector T is a right angle. Use your protractor to check.
Question 2
2
2
2
2.1 R = 500 + 2000 .
R=
R=
 R = 2061.6 newtons
2.2 This is a right-angled triangle. The lift vector L is vertical and the thrust vector T is horizontal.
Question 3
3.1 The sketch should just show the directions of the strings. The strings may be any length
3.2 The sketch will show three vectors the same length. They are equal in length because the tension
force in each string is 3 N.
3.3 The forces in all directions are equal – each force is 3 N. If one force were bigger, it would move
the knot in the direction of the bigger force and the angle between the remaining two forces would
become smaller.
3.4 The vectors will be 3 cm long and at 120° to each other.
3.5. Resultant must be 3 N
upward.
3.6. Vector F1 can be moved
parallel to itself. Adding vector F1 to
vector F2 gives the resultant R
3.7. F1, F2 and F3 can be
added head to tail and they
form a closed polygon.
3.5
3.6
3.7
3.8
The resultant will be 3 cm long and point upwards.
Move vector F1 parallel to itself.
The polygon will be an equilateral triangle. Each interior angle will be 60°.
The angles must be 120° because there are only three angles, they are equal, and they must
share 360° equally between them.
Question 4
4.1 The student’s sketch will look much like the sketch in the video. The purpose of redrawing it is to
become familiar with the situation and the quantities that are given.
4.2
= tan 10°
10 = H tan 10°
=H
H=
H = 56,7 N. The answer must be in newtons because H is a force.
4.3 56,7 N is almost six times the 10 N half-weight of the towel. (The towel weighs 20 N but each half
of the wire holds up half the weight. That is why the diagram shows the upward force on the towel
as 10 N + 10 N.)
4.4
= tan 5°
10 = H tan 5°
=H
H=
H = 11,3 N. So the answer is no, if you want to halve the angle between the wire and the
horizontal, then you must increase the horizontal force component from 56,7 N to 114,3 N, which
is more than double 56,7 N. So when you stretch a washing-line very tightly between the poles,
you will put a very big force on the poles as soon as you put wet washing on the line. The force
will be very much bigger than the weight of the washing.
4.5
= sin 10°
10 = T sin10°
=T
T=
T = 57,5 N The answer must be in newtons because T is a force.
You could have used
= cos 10° but then if you’d calculated H incorrectly, your answer for T
would also be incorrect. You were given the information that the downward component is 10 N,
so use that instead.
Also do a common-sense check on your answer. When you look at the diagram, ask yourself if
your answer for vector T could ever be shorter than the length of vector H?
No, it could not be shorter, because T is the hypotenuse of the triangle, and the hypotenuse is
always longer than the other two sides.
Acknowledgements
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Credits
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